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\begin{document}
\title{The Curve and $p$-adic Hodge theory}
\author{Laurent Fargues}
\date{Recorded by Shenxing Zhang. Not revised yet.}
\maketitle
\tableofcontents
\env{abstract}{The main theme of this course will be to understand and give a meaning to the notion of a $p$-adic Hodge structure. Starting with the work of Fontaine, who introduced many of the basic notions in the domain, it took many years to understand the exact definition of a p-adic Hodge structure. We now have the right definition: this involves the fundamental curve of $p$-adic Hodge theory and vector bundles on it. In the course I will explain the construction and basic properties of the curve. I will moreover explain the proof of the classification of vector bundles theorem on the curve. As an application I will explain the proof of weakly admissible implies admissible. In the meanwhile I will review many objects that show up in $p$-adic Hodge theory like $p$-divisible groups and their moduli spaces, Hodge-Tate and de Rham period morphisms, and filtered $\varphi$-modules.

This is a note of the lectures in MCM, Beijing from 2019/11/01 to 2020/01/10.}

\section{Introduction}
\subsection{What is a $p$-adic Hodge structure?}
Recall a {\em real pure Hodge structure} of weight $w\in\BZ$ is a finitely dimensional real vector space $V$, endowed with a bigrading
\[V_\BC=\bigoplus_{p+q=w}V^{p,q}\]
such that $\ov{V^{p,q}}=V^{q,p}$. For example, let $X/\BC$ be a proper smooth algebraic variety. Then $\RH^i(X(\BC),\BR)$ is equipped with a real Hodge structure of weight $i$ as
\[\RH^i(X(\BC),\BR)_\BC=\bigoplus_{p+q=i}\RH^q(X,\Omega^p).\]



In $p$-adic setting, there are plenty of different structures and results
\env{itemize}{
\item Hodge-Tate Galois representations;
\item crystalline representations;
\item de Rham representations;
\item filtered $\varp$-modules \`a la Fontaine;
\item Breuil-Kisin modules;
\item $(\varp,\Gamma)$-modules;
\item comparison theorems for proper smooth algebraic variety over $\BQ_p$.
}
This is a mess! We should back to real case to find the solution.

\subsection{Real Hodge structure}
Recall Simpson's geometric point of view of twists. Denote
\[\wt{\BP}_\BR^1=\BP_\BC^1/\set{z\sim -\frac{1}{\bar z}}\]
where $z$ is the coordinate on $\BP_\BC^1$. This is a conic curve without real point, equipped with $\infty$. Obviouly $\BP_\BC^1$ is a double cover of $\wt\BP_\BR^1$.
\[\xymatrix{
\BP_\BC^1\ar@{-}[d]_{\BZ/2\BZ}^\pi & 0\ar@{-}[rd]& &\infty\ar@{-}[ld]\\
\wt\BP_\BR^1 & &\infty
}\]

The action of $\BC^\times$ on $\BP_\BC^1$ as $\lambda.z=\lambda z$ descends to an action of $U(1)$ on $\wt\BP_\BR^1$. Then $\infty$ is the unique fixed point of this action and the unqiue point that has a finite orbit.

Consider the vector bundles on $\wt\BP_\BR^1$.
For $\lambda\in\half\BZ$, define
\[\CO_{\wt\BP_\BR^1}(\lambda)=\begin{cases}
\pi_*\CO_{\BP_\BC^1}(2\lambda),\quad &\lambda\notin\BZ;\\
\CL\text{ such that }\pi^*\CL=\CO_{\BP_\BC^1}(2\lambda),&\lambda\in\BZ.
\end{cases}\]
Here the {\em slope} of $\CO_{\wt\BP_\BR^1}(\lambda)$ is $\lambda$.

\env{prop}{
There is a bijection between the set of finite decreasing half integer sequences
\[\set{\lambda_1\ge\cdots\ge \lambda_n\mid \lambda_i\in\half\BZ,n\in\BN}\]
and the isomorphic classes of vector bundles on $\wt\BP_\BR^1$ as
\[(\lambda_i)\longmapsto \left[\bigoplus_i\CO_{\wt\BP_\BR^1}(\lambda_i)\right].\]
In particular,
\[\begin{split}
\cVect_\BR&\simto \set{\text{slope $0$ semisimple vector bundles over }\wt\BP_\BR^1}\\
V&\longmapsto V\otimes \CO_{\wt\BP_\BR^1}\\
\RH^0(\wt\BP_\BR^1,\CE)&\longmapsfrom \CE.
\end{split}\]
}
That is to say, every Harder-Narasimhan filtration of vector bundles are split and every semisimple vector bundle of pure slope are $\CO_{\wt\BP_\BR^1}(\lambda)^n$.

Let $V$ be a real vector space with a filtration $\Fil^\bullet$ on $V_\BC=V\otimes_\BR \BC$. Denote by $t$ the uniformization of $\wt\BP_\BR^1$ at $\infty$ and 
\[V_\BC((t))=V\otimes_\BR \BC((t))=V_\BC\otimes_\BC\BC((t)).\] There is a canonical filtration $\set{t^k\BC[[t]]}_k$ on $\BC((t))$, which induces a filtration on $V_\BC((t))$ as
\[\Fil^k (V_\BC((t)))=\sum_{i\in\BZ} \Fil^i V_\BC\otimes_\BC t^{k-i} \BC[[t]].\]
Then 
\[\wh\CO_{\wt\BP_\BR^1,\infty}=\BC[[t]],\quad (V\otimes_\BR \CO_{\wt\BP_\BR^1})_\infty^\wedge=V_\BC((t))\]
and the $\BC[[t]]$-lattice 
\[\Lambda:=\Fil^0(V_\BC((t)))\subset V_\BC((t))\]
defines a {\em modification} of vector bundles
\[(V\otimes_\BR\CO_{\wt\BP_\BR^1})|_{\wt\BP_\BR^1\bs \set\infty}\simto \CE|_{\wt\BP_\BR^1\bs\set\infty},\]
such that $\wh\CE_\infty=\Lambda.$ This is $U(1)$-equivalent and induces a bijection 
\[\set{\text{filtrations on $V_\BC$}}\simto
\set{\text{$U(1)$-equiv. modif. } V\otimes_\BR\CO_{\wt\BP_\BR^1}\rightsquigarrow\CE}\]
and thus 
\[\set{(V,\Fil^\bullet  V_\BC)}\simto
\set{\text{$U(1)$-equiv. modif. } \CE_1\rightsquigarrow\CE_2 \atop\text{$\CE_1$ semisimple of slope $0$, $U(1)\curvearrowright \RH^0(\CE_1)$ trivially}}.\]

\env{defn}{
A {\em real Hodge structure} is a finitely dimensional real vector space $V$, endowed with a bigrading decomposition 
\[V_\BC=\bigoplus_{p,q\in\BZ}V_\BC^{p,q},\]
such that $\ov{V^{p,q}}=V^{q,p}$.
Thus for any integer $w$, there is a subspace $V_w\subset V$ such that 
\[V_{w,\BC}=\bigoplus_{p+q=w} V^{p,q},\]
which is called {\em weight $w$} part of $V$.
If $V=V_w$, $V$ is called {\em pure of weight $w$}.

We say $(V,\Fil^\bullet V_\BC)$ defines a Hodge struture of weight $w$ if there is a real Hodge struture on $V$ of pure weight $w$ such that $\Fil^n\BV_\BC=\oplus_{p\ge n} V^{p,w-p}$.
}

\env{prop}{
$(V,\Fil^\bullet V_\BC)$ defines a weight $w$ Hodge struture if and only if $\CE_2$ is semisimple of slope $w/2$ in the corresponding modification.
}
This induces a bijection between the set of weight $w$ pure real Hodge structures and the set of $U(1)$-equivalent modifications $\CE_1\rightsquigarrow\CE_2$ on $\wt\BP_\BR^1\bs\set\infty$, where $\CE_1$ is semisimple of slope $0$, $\CE_2$ is semisimple of slope $w/2$ and $U(1)$ acts on $\RH^0(\CE_1)$ trivially.

WE are going to do the same in the $p$-adic setting.
\env{center}{
\begin{tabular}{|c|c|}
\hline real setting&$p$-adic setting\\
\hline $\wt\BP_\BR^1\bs\set\infty\curvearrowleft U(1)$& the curve $X\curvearrowleft \Gal(\bar\BQ_p/\BQ_p)$\\
\hline $\BC[[t]]=\wh\CO_{\wt\BP_\BR^1}$&$B_\dR^+=\wh\CO_{X,\infty}$\\
\hline $ \lambda.t=\lambda t$& $\sigma.t=\chi_\cyc(\sigma)t,\ t=\log[\epsilon]$\\
\hline $\xymatrix{
\BP_\BC^1\ar[d]\ar@/^/@{-}[d]^{\BZ/2\BZ}\\
\wt\BP_\BR^1
}$&
$\xymatrix{
X_\infty\ar[d]\ar@/^/@{-}[d]^{\wh\BZ}\\
X
}$\\
\hline
\end{tabular}
}

Thus the vector bundles on $X$ is endowed with $\Gal(\ov\BQ_p/\BQ_p)$-action.

\section{The curve $Y$}
There are two versions of the curve.
\env{itemize}{
\item $X^\ad$ adic version analog of $p$-adic Reimann surface,
\item $X$ schematical version analog of a proper smooth algebraic curve.
}
There is an analytification morphism (GAGA)  $X^\ad\to X$ and an ``ample'' line bundle $\CO(1)$ on $X^\ad$ such that
\[X=\Proj(\bigoplus_{d\ge0}\RH^0(X^\ad,\CO(d))).\]
Both rely on the construction of an intermediate adic space $Y$ endowed with a ``crystalline'' Frobenius $\varphi$.

Let $C$ be a complete algebraically closed field of characteristic $0$.
Define the tilt $C^\flat$ the inverse limit of $C$ with respect to Frobenius, which is an algebraically closed field of characteristic $p$.
Let $B_\dR^+$ be the completion of $\BA_{\inf}=W(\CO_{C^\flat})$ with repect to $(p-[p^\flat])$ with quotient field $B_\dR$, $A_\cris$ the completion of divided power of $\BA_{\inf}$ and $B_e=B_\cris^{\varphi=1}$.

The $p$-adic comparison theorems for crystalline/de Rham/\etale\ cohomology lead one to consider the category of pairs $(W_e,W_\dR^+)$ where $W_e$ is a free $B_e$-module and $W_\dR^+$ is a free $B_\dR^+$-module such that
\[B_\dR\otimes_{B_e}W_e=B_\dR\otimes_{B_\dR^+}W_\dR^+.\]
We will construct a curve $X$ such that $B_e=\CO(X-\set\infty),B_\dR^+=\CO_{X,\infty}$. The fundamental exact sequence
\[0\ra \BQ_p\ra B_e\ra B_\dR/B_\dR^+\ra 0\]
tells us the sections.
The category of $(W_e,W_\dR^+)$ corresponds to the category of vector bundles over $X$. Since $B_e=B_\cris^{\varphi=1}$, this suggests 
\[X^\ad=Y^\ad/\varphi^\BZ\]
where $Y^\ad=\Spa(A_{\inf})-(p[p^\flat])$.

In general, let $E$ be a discretely valued non-archemedean field with uniformizer $\pi$ with finite residue field $\BF_q=\CO_E/\pi$.
Let $F/\BF_q$ be a perfectoid field, i.e., a perfect field, complete  with respect to a non-trivial absolute value $|\cdot|:F\to \BR_{\ge 0}$.
We will attach to this data a curve $X_{F,E}/E$.
More generally, we can define ``a family of curves'' 
\[X_S=(X_{k(s)})_{s\in|S|}\]
for perfectoid $S/\BF_q$.
If $G$ is a reductive group over $E$, one can define a stack
\[\Bun_G:S\to \set{\text{$G$-bundles on $X_S$}}.\]
We will study the perverse $\ell$-adic sheaves on $\Bun_G$.


\subsection{Affinoid space and adic space}
Let's recall the definition of adic spaces. This is not a prt of the lectures.
Let $k$ be a nonarchimedean field and $R$ a topological $k$-algebra.
\env{defn}{
\env{enumerate}{
\item If there is a subring $R_0\subset R$ such that $\set{aR_0}_{a\in k^\times}$ forms a basis of open neighborhoods of $0$, it's called a {\em Tate $k$-algebra}.
A subset $M\subset R$ is called {\em bounded} if $M\subset aR_0$ for some $a\in k^\times$.
\item An {\em affinoid $k$-algebra} is a pair $(R,R^+)$ consisting of a Tate $k$-algebra $R$ and open integrally closed subring $R^+\subset R^\circ$.
\item An affinoid $k$-algebra $(R,R^+)$ is said to be {\em tft} if $R$ is a quotient of $k\pair{T_1,\dots,T_n}$ for some $n$ and $R^+=R^\circ$.
}
}

\env{defn}{
Denote by $X=\Spa(R,R^+)$ the set of equivalent classes of continuous valuations on $R$, which is $\le 1$ on $R^+$.
We equip $X$ the topology which has open {\em rational subsets}
\[U\left(\frac{f_1,\dots,f_n}{g}\right)=\set{x\in X\mid |f_i(x)|\le |g(x)|,\forall x\in X}\]
as basis, where $f_1,\dots,f_n$ generates $R$.
}

\env{defn}{
A topological space $X$ is called {\em spectral} if it satisfies the following equivalent properties.
\env{enumerate}{
\item There is some ring $A$ such that $X\cong \Spec A$.
\item $X$ is an inverse limit of finite $T_0$ spaces.
\item $X$ is quasicompact, has a quasicompact topological basis, stable under finite intersections, and every irreducible closed subset has a unique generic point.
}
}

\env{thm}{
The space $\Spa(R,R^+)$ is spectral and $\Spa(R,R^+)\cong \Spa(\wh R,\wh R^+)$.
}
\env{thm}{\env{enumerate}{
\item If $X=\emptyset$, then $\wh R=0$.
\item If $R$ is complete and $|f(x)|\neq 0,\forall x\in X$, then $f$ is invertible.
\item If $|f(x)|\le 1,\forall x\in X$, then $f\in R^+$.
}}

Consider the topological algebra $R[f_1g^{-1},\dots,f_nh^{-1}]\subset R[g^{-1}]$ and denote by $B$ the integral closure of $R^+[f_1g^{-1},\dots,f_ng^{-1}]$ in it, then $(R[f_1g^{-1},\dots,f_ng^{-1}],B)$ is an affinoid $k$-algebra with completion $(R\pair{f_1g^{-1},\dots,f_ng^{-1}},\wh B)$. Then
\[\Spa(R\pair{f_1g^{-1},\dots,f_ng^{-1}},\wh B)\to \Spa(R,R^+)\]
factors through $U\left(\frac{f_1,\dots,f_n}{g}\right)$ and it satisfies the corresponding universal property.
Define presheaves
\[(\CO_X(U),\CO_X^+(U))=(R\pair{f_1g^{-1},\dots,f_ng^{-1}},\wh B)\]
and on general $W$,
\[\CO_X=\plim_{U\subset W}\CO_X(U).\]
Moreover $U\cong \Spa(\CO_X(U),\CO_X^+(U)).$

The stalk $\CO_{X,x}$ is a local ring with maximal ideal $\set{f\mid f(x)=0}$ and $\CO^+_{X,x}$ is a local ring with maximal ideal $\set{f\mid f(x)<1}$.

\env{defn}{
We call $R$ is {\em strongly neotherian} if $\wh R\pair{T_1,\dots,T_n}$ is noetherian for any $n$.
}
\env{thm}{
If $R$ is strongly neotherian, then $\CO_X$ is a sheaf.
}

\env{defn}{
Consider triple $(X,\CO_X,(|\cdot(x)|,x\in X))$ where $(X,\CO_X)$ is a locally ringed space and $|\cdot(x)|$ is a continuous valuation on $\CO_{X,x}$ for any $x\in X$.
Such triple isomophic to $\Spa(R,R^+)$ where $\CO_X$ is a sheaf is called an {affinoid adic space}.

It is called an {\em adic space} if it's locally an affinoid adic space.
}

\env{prop}{
For affinoid adic space $X=\Spa(R,R^+)$ and  any adic space $Y$ over $k$,
\[\Hom(Y,X)=\Hom((\wh R,\wh R^+),(\CO_Y(Y),\CO_Y^+(Y))).\]
}

\env{exam}{
Assume that $k$ is complete and algebraically closed.
Let $R=k\pair{T}$ and $R^+=R^\circ=k^\circ\pair{T}$.
Fix a norm $|\cdot|:k\to\BR_{\ge 0}$.
Then $X=\Spa(R,R^+)$ consists of

\env{center}{
\includegraphics[height=120pt]{adic.png} 
}

(1) The classical point.
For $x\in k^\circ$,
\[\fct{R}{\BR_{\ge0}}{f=\sum a_nT^n}{|f(x)|=|\sum a_n x^n|.}\]


(2)(3) The rays of the tree. For $0\le r\le 1,x \in k^\circ$,
\[\fct{R}{\BR_{\ge0}}{f=\sum a_n(T-x)^n}{\sup |a_n| r^n=\sup _{y\in k^\circ,|y-x|\le r}|f(y)|.}\]
If $r=0$, it is the classical point.
If $r=1$, it doesnot depend on $x$, which is called the Gausspoint.

If $r\in|k^\times|$, it's said to be of type (2), otherwise of type (3).

(4) Dead ends of the tree.
Let $D_1\supset D_2\supset \cdots$ be a sequence of disks with $\cap D_i=\emptyset$. It occurs when $k$ is not spherically complete.
\[\fct{R}{\BR_{\ge0}}{f}{\inf_i \sup _{x\in D_i}|f(x)|.}\]

(5) For $\Gamma=\BR_{\ge 0}\times \gamma^\BZ$, where $\gamma=r^-$ or $r^+ (r<1)$.
\[\fct{R}{\Gamma\cup\set{0}}{f=\sum a_n(T-x)^n}{\sup |a_n|\gamma^n.}\]
This only depends on the disc $D(x,<r)$ or $D(x,r)$.
Thus if $r\notin |k^\times|$, it's of type (3). Every rays of point of type (2) correspond a valuation of type (5).
}



\subsection{Holomorphic function of the variable $p$}

Let $E$ be a finite extension of $\BQ_p$ with residue field $\BF_q$. As a comparison, we also take $E=\BF_q[[t]]$. It is the coefficient field of the $p$-adic Hodge theory.
\env{defn}{
Define
\[\BA=\BA_{\inf}=\begin{cases}
W_{\CO_E}(\CO_F)=W(\CO_F)\otimes_{W(\BF_q)}\CO_E,\quad&E/\BQ_p,\\
\CO_F\wh\otimes_{\BF_q}\CO_E=\CO_F[[\pi]],& E=\BF_q[[t]].
\end{cases}\]
Then
\[\BA=\set{\left.\sum_{n\ge 0}[x_n]\pi^n\right| x_n\in\CO_F}.\]
}

Fix $\varpi\in F$ with $0<|\varpi|<1$. Then $\BA$ is complete under the $(\pi,[\varpi])$-adic topology.
Consider the adic space $\Spa(\BA,\BA)$. It has only one closed point with kernel $(\pi,\fm_F)$.
Define
\[\CY=\Spa(\BA,\BA)_a=\Spa(\BA,\BA)\bs\set{\text{closed point}}=\Spa(\BA,\BA)\bs V(\pi,[\varpi])\]
and an open subspace 
\[Y=\Spa(\BA,\BA)\bs V(\pi[\varpi]).\]
Here the subscript $a$ indicates we take the analytic points and $\CY$ is not affinoid.

Consider the space of holomorphic functions $\CO(Y)$.
Let
\[\BA\left[\frac{1}{\pi},\frac{1}{[\varpi]}\right]=
\set{\left.\sum_{n\gg-\infty}[x_n]\pi^n\right| x_n\in F,\sup|x_n|<+\infty}\]
be the set of holomorphic functions on $Y$ that are meromorphic along $(\pi),([\varpi])$.
For $\rho\in(0,1), f=\sum_{n\gg-\infty}[x_n]\pi^n$, define the Gauss norms
\[|f|_\rho:=\sup_n|x_n|\rho^n=\sup_{|y|\le \rho}f(y).\]

\env{prop}{
The space
\[B=\CO(Y)\]
is the completion of $\BA\left[\frac{1}{\pi},\frac{1}{[\varpi]}\right]$ with respect to $\set{|\cdot|_\rho}$.
}
For compact subset $I\subset (0,1)$, the completion $B_I$ with respect to $\set{|\cdot|_{\rho\in I}}$ is a Banach $E$-algebra and
\[B=\plim_{I\subset(0,1)} B_I\]
is a Fr\'echet space.
In particular, if $I=[\rho_1,\rho_2]$, $B_I$ is the completion with respect to $\set{|\cdot|_{\rho_1},|\cdot|_{\rho_2}}$.

In the case $E=\BF_q[[\pi]]$,
\[Y=\BD_F^*=\set{0<|\pi|< 1}\subset \BA_F^1\]
and
\[B=\CO(Y)=\set{\left.\sum_{n\gg-\infty}x_n\pi^n\right| x_n\in F,\lim_{n\to+\infty}|x_n|\rho^n=0,\forall \rho}.\]
We have natural maps
\[\xymatrix{
&\BD_F^*\ar[ld]\ar[rd]\\
\Spa(F)&&\BD_{\BF_q}^*=\Spa(\BF_q((\pi)))=\Spa(E).
}\]
The map on the left is locally of finite type, but $\BD_F^*\to \Spa(E)$ is not.

\env{remark}{
If $(x_n)\in F^\BZ$ such that $\lim_{|n|\to+\infty}|x_n|\rho^n=0,\forall \rho$, then $\sum [x_n]\pi^n\in B$. But not every element can be written in this form.
}

\subsection{Newton polygon}
\env{prop}{$|fg|_\rho=|f|_\rho|g|_\rho$, i.e., $|\cdot|$ is a valuation.
}
For $\rho=q^{-r},r\in(0,+\infty),|f|_\rho=q^{-v_r(f)}$, where 
\[v_r(f):=\inf(v(x_n)+nr).\]
Here $v=-\log_q|\cdot|$ on $F$. Then $r\mapsto v_r(f)$ is a convex function.

In the case $E=\BF_q[[\pi]]$, $f=\sum x_n\pi^n\in\CO(Y)$ defines a Newton polygon $\Newt(f)$ the decreasing convex hull of $\set{(n,v(x_n))}.$ Then positive slopes of $\Newt(f)$ one-to-one correspond to the set of valuations of roots of $F$ on $\BD_F^*$.

Assume $E/\BQ_p$. Recall the Legendre transform gives a bijection between the set of convex decreasing function $\BR\to\BR\cup\set{\infty},\not\equiv+\infty$ and the set of concave function $(0,+\infty)\to \BR\cup\{-\infty\},\not\equiv-\infty$ as
\[\begin{split}
\CL(\varphi)(r)&=\inf_{t\in\BR}(\varphi(t)+tr),\\
\CL^{-1}(\psi)(t)&=\sup_{r\in(0,\infty)}(\psi(r)-tr).
\end{split}\]

\env{prop}{
For convex decreasing function $f,g:\BR\to \BR\cup\set\infty$, we have
\[\CL(f\circledast g)=\CL(f)+\CL(g),\]
where
\[(f\circledast g)(x)=\inf_{a+b=x}(f(a)+g(b)).\]
}

The Legendre transform maps polygons to polygons, and the slopes of $\varphi$ (resp. $\psi$) one-to-one correspond to the $x$-coordinates of break points of $\CL(\varphi)$ (resp. $\CL^{-1}(\psi)$).

\env{prop}{
For nonzero $f\in B$, there is a sequence $\set{f_n}$ in $\BA\left[\frac{1}{\pi},\frac{1}{[\varpi]}\right]$ tending to $f$. Then for any compact subset $K\subset (0,+\infty)$, there is an integer $N$ such that for any $n\ge N$, $v_r(f)=v_r(f_n)$ for any $r\in K$.

As a corollary, the convex function $r\mapsto v_r(f)$ is a polygon with integral slopes.
}
Define
\[\Newt(f):=\CL^{-1}(r\mapsto v_r(f)).\]
Then
\[\Newt(fg)=\Newt(f)\circledast\Newt(g).\]

Let $I\subset (0,1)$ be a compact subset and $0\neq f\in B_I$. Denote by $\Newt_I(f)$ the part of Newton polygon consisting of the slope in $-\log_q(I)$ part. But $\set{v_r(f)}_{r\in -\log_q(I)}$ do not determine $\Newt_I(f)$. For example, $I=\set{q^{-r}}$, we need to know the left and right break point of the slope $r$ part to determine $\Newt_I(f)$.

Denote by $\partial_l,\partial_r$ the left/right derivation.
Then $(v_r(f),\partial_l v_r(f),\partial_r v_r(f))_{r\in-\log_q (I)}$ determine $\Newt_I(f)$. The rank $2$ valuations with image in $\BR\times \BZ$
\[f\mapsto (v_r(f),-\partial_l v_r(f)),\]
\[f\mapsto (v_r(f),\partial_r v_r(f)),\]
are specializations of $v_r$.

\subsection{Zeros of holomorphic functions}
Recall Jensen's inequality/equality. For nonzero $f\in \CO(\BC)$ such that $f(0)\neq0$. Let $R>0$ such that $f$ has no zero on $\set{|z|=R}$. Let $a_1,\dots,a_n$ be zeros of $f$ in $\set{|z|<R}$.
Then
\[\ln|f(0)|=\frac{1}{2\pi}\int_0^{2\pi}\ln|f(Re^{i\theta})|\rmd \theta-n\ln R+\sum_{i=1}^n\ln|a_i|\]
and
\[\ln|f(0)|\le M(R)-n\ln R+\sum_{i=1}^n\ln|a_i|,\]
where $M(R)$ is the maximal modulus on $\set{|z|=R}$.

In the non-zrchimedead setting, there is  an equality.
Assume $E=\BF_q[[\pi]]$. For nonzero $f=\sum_{n\ge 0}x_n\pi^n\in\CO(\BD_F)$, $f(0)=x_0\neq 0$. Assume it has roots $(a_i)_{i\ge1}$ with $v(a_1)\ge v(a_2)\ge \dots$. Then the slopes of $\Newt(f)$ are valuations of roots of $f$,
\[v(f(0))=v_r(f)-nr+\sum_{i=1}^n v(a_i).\]
We want to do the same for $E=\BQ_p$. We need to define zeros of $f$ in this setting.

For $E=\BF_q((\pi))$,
\[Y=\BD_F^*=\set{0<|\pi|<1}\]
and
\[\begin{split}
|Y|^\cl&=\set{z\in\bar F\mid 0<|z|<1}/\Gal(\bar F/F)\\
&=\set{P\in F[\pi]\mid \text{irreducible with all roots such that }0<|z|<1}/F^\times\\
&=\set{P\in\CO_F[\pi]\mid \text{unitary irreducible such that }0<|P(0)|<1}.
\end{split}\]

\env{defn}{
$f=\sum_{n\ge 0}x_n \pi^n\in \BA$ is (distinguished) primitive of degree $d>0$ if $x_0\neq0, x_0,\dots,x_{d-1}\in\fm_F,x_d\in\CO_F^\times.$
}
By Weierstrass fatorization, $f=uP$ uniquely where $u\in\CO_F[[\pi]]^\times$ and $P\in \CO_F[\pi]$ is unitary with degree $d$. Thus
\[|Y|^\cl=\set{\text{primitive irreducible elements}}/\CO_F[[\pi]]^\times.\]

Assume $E/\BQ_p$.
\env{defn}{$f=\sum_{n\ge 0}[x_n]\pi^n\in\BA$ is primitive of degree $d$ if $x_0\neq 0, x_0,\dots,x_{d-1}\in\fm_F,x_d\in\CO_F^\times$. 
}
It's equivalent to say, $f\mod \pi\neq 0$ in $\CO_F$ and $f\mod W_{\CO_E}(\CO_F)\neq0$ in $W_{\CO_E}(k_F)^d$.
The degree of $f$ is $v_\pi(f\mod W_{\CO_E}(\CO_F))$.
Thus $\deg(fg)=\deg f+\deg g.$

\env{defn}{
\[|Y|^\cl=\set{\text{irreducible primitive}}/\BA^\times.\]
}
We will show that this is the set of the classical points of $Y$.

\subsection{Perfectoid fields and tilting}
\env{defn}{
A complete field $K$ with respect to a norm $|\cdot|: K\to\BR_{\ge 0}$ is called a {perfectoid field}, if there is an element $\varpi\in K$ such that $|p|\le|\varpi|<1$ such that $\Frob:\CO_K/\varpi\to \CO_K/\varpi$ is surjective.
}
For example, $\wh{\BQ(\zeta_{p^\infty})} (p>2),\wh{\BQ_p(p^{1/p^\infty})}$. An algebraic closed complete valued field is perfectoid. In char $p$ case, $K$ is perfectoid if and only if it is perfect.

Let $K$ be a perfectoid field. Define the {\em tilting}
\[K^\flat=\plim_{x\mapsto x^p} K=\set{(x^{(n)})_{n\ge 0}\in K^\BN\mid (x^{(n+1)})^p=x^{(n)}},\]
with
\[(xy)^{(n)}=x^{(n)}y^{(n)},\quad (x+y)^{(n)}=\lim_{k\to +\infty} (x^{(n+k)}+y^{(n+k)})^{p^k}.\]
Define
\[x^\#:=x^{(0)}\]
and
\[\fct{|\cdot|:K^\flat}{\BR_{\ge 0}}{x}{|x^\#|.}\]
Then $K^\flat$ is also perfectoid.
Moreover, there is an isomorphism
\[\begin{split}
\CO_{K^\flat}&\simto \plim_{x\mapsto x^p}\CO_K/p\\
x&\mapsto (x^{(n)}\mod p)_{n\ge0}\\
\lim_{k\to +\infty}(\hat y_{n+k})^{p^k}&\mapsfrom (y_n)_{n\ge0}.
\end{split}\]

\env{exam}{If $K$ is of characteristic $p$, then $K^\flat=K$.}
\env{exam}{
If $K=\wh{\BQ_p(\zeta_{p^\infty})}, \epsilon=(\zeta_{p^n})_{n\ge0}\in K^\flat$ and $\pi_\epsilon=\epsilon-1\in K^\flat$, then $K^\flat=\BF_p((\pi_\epsilon^{1/p^\infty})).$ In fact, $\BZ_p(\zeta_{p^\infty})/p\simto \BF_p(\pi_\epsilon^{1/p^\infty})/\pi_\epsilon$.

If $K=\wh{\BQ_p(p^{1/p^\infty})}, \pi=(p^{1/p^n})_{n\ge 0}\in K^\flat$, then $K^\flat=\BF_p((\pi^{1/p^\infty})).$ In fact, $\BZ_p(p^{1/p^\infty})/p\simto \BF_p(\pi^{1/p^\infty})/\pi$.
}

\env{rem}{In fact, Fontaine gave the isomorphism
\[R^\flat=\plim_{x\mapsto x^p}R/pR \simto \set{(x^{(n)})_{n\ge 0}\in R^\BN\mid (x^{(n+1)})^p=x^{(n)}}\]
for any separated complete $p$-adic ring $R$.
}

\env{thm}{Let $K$ be a perfectoid field. Then
\env{enumerate}{
\item If $L/K$ is finite, then $L$ is perfectoid and $[L^\flat:K^\flat]=[L:K]$.
\item $\CO_L/\CO_K$ is almost \etale, i.e., if $n=[L:K], \forall 0<\epsilon<1, \exists e_1,\dots,e_n\in \CO_L$ such that 
\[\epsilon\le|\disc(\Tr_{L/K}(e_ie_j))_{1\le i,j,\le n}|\le1.\]
\item $(\cdot)^\flat$ induces an equivalence between the set of finite \etale\ $K$-algebras and the set of finite \etale\ $K^\flat$-algebras.
}}
\env{coro}{
\env{enumerate}{
\item $K$ is algebraically closed if and only if $K^\flat$ is.
\item $\Gal(\ov K/K)\simto \Gal(\ov{ K^\flat}/K^\flat)$, where $\ov{ K^\flat}$ is the union of all $L^\flat$ where $L/K$ is finite.
}}
\env{prop}{
The functors
\[\xymatrix{\set{\text{$p$-adic rings}}
\ar@<1ex>[r]^-{(\cdot)^\flat}
&\set{\text{perfect $\BF_p$-algebras}}
\ar@<1ex>[l]^-{W(\cdot)}
}\]
are adjoint, i.e.,
\[\Hom(W(A),B)=\Hom(A,B^\flat).\]
The adjuncation morphisms are
\[\begin{split}
R&\simto W(R^\flat)\\
x&\mapsto [x^{1/p^n}],\end{split}\]
\[\begin{split}
\theta: W(R^\flat)&\simto R\\
\sum[x_n]p^n&\mapsto \sum x_n^\# p^n.\end{split}\]
}

\env{rem}{If $R$ is a $p$-adic ring such that the Frobenius on $R/pR$ is surjective, then $\theta\mod p$ is $R^\flat\to R/pR$. Thus $\theta$ is surjective by Nakayama lemma and $R$ is a quotient of $W(R^\flat)$.
}
\subsection{Classical points}
\env{thm}{
Let $\xi$ be an irreducible primitive element of degree $d$ and $\theta:\BA\surj \BA/\xi=\CO_K,K=\CO_K[1/p]$.
\env{enumerate}{
\item $K/E$ is a perfectoid field with $|\theta([x])|=|x|.$
\item The morphism
\[\fct{\CO_F}{\CO_K^\flat}{x}{\theta([x^{p^{-n}}])_{n\ge0}}\]
induces $K^\flat/F$ of degree $d$. In particular, $K^\flat=F$ if $d=1$.
\item For $d=1$, this induces 
\[\begin{split}
|Y|^{\cl,\deg=1}=\mathrm{Prim}^{\deg=1}/\BA^\times&\simto \set{K/E\text{ perfectoid }, K^\flat=F}/\sim\\
(\xi)&\mapsto(\BA/\xi)[1/p]\\
\ker\theta&\mapsfrom K/E.
\end{split}\]
}
}
Thus any $\xi$ defines a valuation
\[\BA\left[\frac{1}{\pi},\frac{1}{[\varpi]}\right]
\to \BA\left[\frac{1}{\pi},\frac{1}{[\varpi]}\right]/\xi\sto{|\cdot|}\BR_{\ge0},\]
and
\[|Y|^\cl=\set{V(\xi)\mid \xi\in\BA\text{ irreducible primitive }}\subset|Y|.\]
We see that for $y\in|Y|^\cl$, $k(y)/E$ is perfectoid and $[k(y)^\flat:F]<+\infty$.

\env{thm}{
Assume that $F$ is algebraically closed.
\env{enumerate}{
\item $\forall y\in|Y|^\cl,k(y)$ is algebraically closed.
\item $\forall \xi,\deg(\xi)=1$.
\item any primitive element $\xi$ can be written as
\[\xi=u(\pi-[a_1])\cdots(\pi-[a_d])\]
where $u\in\BA^\times$.
}
}

For $y=V(\xi)\in|Y|^\cl,\xi=\sum[x_n]\pi^n$ is primitive of degree $d$, set
\[|\xi|=|x_0|^{1/d}=|\pi(y)|.\]
This defines the radius
\[|\cdot|:|Y|^\cl\to(0,1).\]
\env{defn}{
For $y=V(\xi)\in|Y|^\cl,$
\[B_{\dR,y}^+=\xi\text{-adic completion of }\BA\left[\frac{1}{\pi},\frac{1}{[\varpi]}\right]=\wh{\CO_{Y,y}}.\]
It is a discrete valuation ring with uniformizer $\xi$ and residue field $k(y).$
}

\subsection{Localization of zeros}
\env{thm}{
For nonzero $f\in B$,
\[\set{-\log_q|y|\mid y\in|Y|^\cl,f(y)=0}\]
coincides the slopes of $\Newt(f).$
}
\env{defn}{For any interval $I\subset(0,1)$, 
\[|Y_I|^\cl=\set{y\in|Y|^\cl\mid |y|\in I}.\]
}
\env{thm}{For any compact subset $I\subset(0,1)$, $B_I$ is a PID with $\Spm B_I=|Y_I|^\cl.$ In fact, $\Spm B=\set{(\xi)\mid |\xi|\in I}.$
}
\env{prop}{
\[B_I^\times=\set{f\in B_I\bs\set0 \mid \Newt(f)=\emptyset}.\]
}

Define the {\em Robba ring} the local ring of $\CY$ 
at origin,
\[\CR=\plim_{\rho\to 0^+} B_{(0,\rho]}.\]
This is a Bezout ring.

Define
\[\Div^+(Y_I)=\{D=\sum_{y\in|Y_I|^\cl} m_y[y]\mid \supp(D)\text{ is locally finite },m_y\in\BN\},\]
and
\[\fct{\div: (B_I\bs\set0)/B_I^\times}{\Div^+(Y_I)}{f}{\sum \ord_y(f)[y].}\]
\env{rem}{
If $E=\BF_q((\pi)), I=(0,1)$, the div map is a bijection if and only if $F$ is spherically complete (Larzard).
}

For any $\rho\in(0,1),$
\[\div: B_{(0,\rho]}\bs\set0/B_{(0,\rho]}^\times\simto \Div^+(Y_{(0,\rho]}).\]
In fact, for $D=\sum_{n\ge 0} [y_n]$ with $|y_n|\to 0$, write $y_n=V(\xi_n),$ the series
\[f=\prod_{n\ge0}\xi_n\pi^{-\deg \xi_n}\]
converges, where $\xi_n\equiv \pi^{\deg\xi}\mod W_{\CO_E}(\CO_F)$.


\subsection{Parametrization of classical points}
Assume $F$ is algebraically closed.
If $E=\BF_q((\pi))$, then $|Y|^\cl=|D_F^*|^\cl=\fm_F\bs\set0$. Thus
\[\begin{split}
D^*(F)=\fm_F\bs\set0&\stackrel{\sim}{\lra} |Y|^\cl\\
a&\longmapsto V(\pi-a).
\end{split}\]


If $E/\BQ_p$, $a\in\fm_F\bs\set0, y=V(\pi-[a])$, 
\[\begin{split}
D^*(F)=\fm_F\bs\set0&\relbar\joinrel\twoheadrightarrow |Y|^\cl\\
a&\longmapsto V(\pi-a).
\end{split}\]
But it's hard to describe fibers.

For $y\in|Y|^\cl, C_y=k(y)/E$ is algebraically closed.
Choose $\ul\pi\in C_y^\flat$ such that $\ul\pi^\sharp=\pi$. Then $y=V(\pi-[\ul\pi]).$

Consider the case $E=\BQ_p$. It's same for general $E$ by using Lubin-Tate groups.
Then 
\[\hGm(\CO_F)=(1+\fm_F,\times)\]
is a Banach space as
\[a. \epsilon=\sum_{k\ge 0}{a\choose k} (\epsilon-1)^k,\]
\[p. \epsilon=\epsilon^p\]
and the fact that $F$ is perfect.
\env{defn}{
For any $1\neq \epsilon\in1+\fm_F$,
\[u_\epsilon:=\frac{[\epsilon]-1}{[\epsilon^{1/p}]-1}=1+[\epsilon^{1/p}]+\cdots+[\epsilon^{\frac{p-1}{p}}]\in\BA.\]
}
\env{lem}{
$u_\epsilon$ is primitive of degree $1$.
}
Indeed, 
\[u_\epsilon\mod p=1+\epsilon^{1/p}+\cdots+\epsilon^{(p-1)/p}=\frac{\epsilon-1}{\epsilon^{1/p}-1}\in \CO_F\]
is nonzero,
\[u_\epsilon \mod W(\fm_F)\equiv 1+[1]+\cdots+[1]=p\in W(k_F).\]

Set
\[C_\epsilon=B/u_\epsilon=k(y)\]
where $y=V(\epsilon)$.
Then $\epsilon=(\epsilon^{(n)})\in F=C_\epsilon^\flat$, where $\epsilon^{(n)}=\theta_\epsilon([\epsilon^{1/p^n}])$.
Then
\[1+\epsilon^{(1)}+\cdots+(\epsilon^{(1)})^{p-1}=\theta_\epsilon(1+[\epsilon^{1/p}]+\cdots+[\epsilon^{(p-1)/p}])=\theta_\epsilon(u_\epsilon)=0,\]
thus $\epsilon^{(1)}\in\mu_p(C_\epsilon).$
Moreover
\[\CO_{C_\epsilon}/p\CO_{C_\epsilon}=\BA/(p,u_\epsilon)=\CO_F/\bar u_\epsilon,\]
where
\[\bar u_\epsilon=\frac{\epsilon-1}{\epsilon^{1/p}-1}=(\epsilon-1)^{\frac{p-1}{p}}.\]
Since $\epsilon^{(1)}-1\equiv \epsilon^{1/p}-1\mod p$, $\epsilon^{1/p}-1\notin \CO_F u_\epsilon,$ $\epsilon^{(1)}-1\neq 0\mod p$ in $C_\epsilon$.
Hence $\epsilon^{(1)}\in\mu_p(C_\epsilon)$ is primitive and $\ul\epsilon$ is a generator of $\BZ_p(1)(C_\epsilon)=\set{x\in C_\epsilon^\flat\mid x^\sharp=1}.$

\env{prop}{
\[\begin{split}
((1+\fm_F)\bs\set1)/\BZ_p^\times&\stackrel{\sim}{\longrightarrow} |Y|^\cl\\
\epsilon&\lto V(u_\epsilon).
\end{split}\]
}
The inverse is given by $y\in|Y|^\cl,C_y=k(y)/E$.
Choose $\epsilon$ a basis of $\BZ_p(1)(C_y)\inj (C_y^\flat)^\times=F^\times.$ Then $\epsilon\in(1+\fm_F)\bs\set1,y=V(u_\epsilon).$

\env{remark}{
Let
\[Y^\diamond=\Spa F \times_{\Spa \BF_p} (\Spa \BQ_p)^\diamond)=\Spa F\times \Spa \BQ_p^{\cyc,\flat}/\BZ_p^\times=\BD_F^{*,1/p^\infty}/\BZ_p^\times.\]
where $\BZ_p^\times=\Gal(\BQ_p^\cyc/\BQ_p),$
 $\BQ_p^{\cyc,\flat}=\BF_p((T^{1/p^\infty})).$
 The action of $\BZ_p^\times$ is given by $a.T=\sum_{k\ge 0}{a\choose k}(T-1)^k.$
  Then $|Y|=|Y^\diamond|=|\BD_F^*|/\BZ_p^\times.$
  \[|Y_{\hat{\bar F}}|^{\cl,G_F\text{-finite}}
  \surj |Y_F|^\cl\]
and $|Y_F|^\cl=|Y_{\hat{\bar F}}|^{\cl,G_F\text{-finite}}/G_F=((1+\fm_{\hat{\bar F}})\bs\set1)/\BZ_p^\times)^{G_F\text{-finite}}/G_F$.
}


\section{The curve $X$}
The curve $Y$ is Stein, it's completely determined by the $E$-Frech\'et algebra $\CO(Y)$. It's {\em preperfectoid}, i.e., $Y\wh\otimes_E K$ is perfectoid for a perfectoid field $K/E$.
The Frobenius $\varphi$ acts on $\BA$ by
\[\sum[x_n]\pi^n\mapsto \sum[x_n^q]\pi^n.\]
This induces the action of $\varphi$ on $\CO(Y)$ and $Y$ with $|\varphi(y)|=|y|^{1/q}$.
\env{thm}{
(1) $Y\pair{\frac{\pi^a}{[\varpi]^b},\frac{[\varpi]^c}{\pi^d}}=\Spa(R,R^\circ)$  and $R$ is an $E$-Banach algebra and a PID.

(2)  $R$ is strongly neotherian.
}
Thus $Y$ is a one-dimensonal regular adic space over $E$.
Define
\[X^\ad:=Y/\varphi^\BZ,\]
this is a quasi-compact adic space over $E$, neotherian regular of dimension one. 
For $0<\rho_1<\rho_2<\rho_1^{1/q}<1,$
\[X^\ad=Y_{[\rho_1,\rho_2]}\cup Y_{[\rho_2,\rho_1^{1/q}]}.\]

\env{remark}{
$\varp$ is the arithmetic Frobenius. For $X/\BF_q$, there are geometric Frobenius $\Frob_X\times\Id$, arithmetic Frobenius $\Id\times \Frob_q$ and absolute Frobenius $\Frob_X\times \Frob_q$ on $X_{\bar\BF_q}$.
}
The line bundles on $X^\ad$ are $\varphi^\BZ$-equivariant line bundles over $Y$, i.e., projective $\varphi$-modules over $B$ of rank $1$, or free $\CR$-modules of rank $1$.
Thus $\Pic(X^\ad)=\BZ$, where $n$ corresponds $(B\cdot e,\varphi)$ with $\varphi(e)=\pi^{-n}e$.
\env{defn}{
Define $\CO(d)$ corresponds $(B,\pi^{-d}\varphi)$.
}

For a proper smooth algebraic curve $X$ over $\BC$, the analytic part $X^\an$ is a compact Riemann surface. Conversely, given a compact Riemann surface $Z$, there is an ample line bundle $\CL$ over $Z$, e.g., $\CO(z)$ for $z\in Z$, Then
\[\Proj\bigl(\bigoplus_{d\ge 0}\RH^0(Z,\CL^{\otimes d})\bigr)\]
is a proper smooth algebraic curve.

We claim that $\CO(1)$ is ample.
Denote by 
\[P_d=\RH^0(X^\ad,\CO(d))=B^{\varp=\pi^d}\]
and
\[P=\bigoplus_{d\ge 0}P_d.\]
We take
\[X=\Proj P.\]

\env{thm}{
(1) $X$ is a Dedekind scheme.

(2) There is a natural morphism of ringed spaces $X^\ad\to X$ inducing $|X^\ad|^\cl:=|Y|^\cl/\varp^\BZ\simto |X|=\set{\text{closed points}}$ such that 
\[\wh\CO_{X,x}\simto \wh\CO_{Y,y}=B_\dR^+(k(y))\]
if $y\mapsto x$. In particular, for any $x\in|X|$, $k(x)/E$ is perfectoid.

(3) $X$ is complete, i.e., for any $x\in|X|$, $\deg(x):=[k(x)^\flat:F]$, then $\deg(\div(f))=0$ for any $f\in E(X)^\times$. This implies we may define degree of vector bundles.

(4) There is an isomorphism 
\[\fct{|X|^{\deg=1}}{\set{\text{untilts of $F$}}/\Frob^\BZ}{x}{k(x).}\]

(5) If $F$ is ac, $\infty\in|X|$, there is $t\in \RH^0(X,\CO(1))\bs\set0$ such that $V(t)=\set\infty$ and $X\bs\set\infty=\Spec B_e$, where $B_e:=B[1/t]^{\varphi=1}$.
}
$B_e$ is a PID and $(B_e,-\ord_\infty)$ is non-Euclidean but almost Euclidean, i.e., for any $x,y$, there is $x=ay+b$ with $\deg(b)\le \deg (y)$. That's because $\RH^1(X,\CO_X(-1))\neq 0$ but $\RH^1(X,\CO_X)=0$.

We are going to prove that $X$ is a curve. We assume that $F$ is algebraically closed. The case of general perfectoid $F$ is treated by Galois descent from $\wh F$ to $F$.

\subsection{The fundamanetal exact sequence}

\env{prop}{
$P$ is a graded fractional ring with irreducible elements of degree $1$.
}
For any $0\neq t\in P_1$, $P[1/t]_0$ is fractional with irreducible elemenets $\set{t'/t\mid t'\in P_1-Et}.$

\env{prop}{
Let $t_1,\dots,t_d\in P_1\bs\set0$ associate $y_1,\dots,y_d\in|Y|^\cl$, i.e., $\div(t_i)=\sum_{n\in\BZ} [\varp^n(y_i)]$ on $Y$.
Let $y_i=V(a_i)$, where $a_i$ is primitive of degree $1$.
Then the sequence
\[0\ra E\cdot\prod_{i=1}^d t_i\ra B^{\varp=\pi^d}\ra B/a_1\cdots a_d B\ra 0\]
is exact.
}
For example, if $t\in P_1\bs\set0$, 
\[0\ra E\cdot t^d\ra B^{\varp=\pi^d}\ra B_{\dR,y}^+/\Fil^d B_{\dR,y}^+\ra 0\]
for $y\in|Y|^\cl, t(y)=0$.
\env{proof}{
Exactness in the middle. Suppose $f\in B^{\varp=\pi^d}\cap a_1\dots a_d B$ is nonzero,  then
\[\div(f)\ge \sum_{i=1}^d[y_i]\]
in $\Div^+(X^\ad).$ Then
\[\div(f)\ge \sum_{n\in\BZ} \sum_{i=1}^d [\varp^n(y_i)]=\div(\prod_{i=1}^d t_i)\]
and then $f=x\prod\limits_{i=1}^d t_i$ for some $x\in B^{\varp=1}=E$.

Surjectivity. We only need to prove $d=1$ case. For any $x\in C$, $p^n x$ lies in the convergence domain of $\exp$ for $n\gg0$. Since $C$ is algebraically closed, there is $z\in C$ such that $\exp(p^n x)=z^{p^n}$, thus $\log z=x$ and $\log:1+\fm_C\to C$ is surjective.

Assume $E=\BQ_p$.
Let $a$ be a primitive element of degree $1$, $y=V(a)$ and $C=C_y=B/aB$. Then $C^\flat=F$. For any $\varepsilon\in 1+\fm_F$, $\log([\varepsilon])\in B^{\varp=p}$ and $\theta(\log([\varepsilon]))=\log(\theta([\varepsilon]))=\log \varepsilon^\sharp$, $\varepsilon^\sharp\in 1+\fm_C$. Take $\varepsilon$ such that $\varepsilon^\sharp=z$, then $\theta(\log([\varepsilon]))=x$.
It's same for general $E$ by using $\log_\LT$ for Lubin-Tate group with respect to $(q,\pi)$.
}

We are going to use the fundamental exact sequence to prove that $X$ is a curve. Reciprocally, once the curve is constructed, we can find back the fundamental exact sequence by applying $\RH^0(X,-)$ on
\[\xymatrix{0\ar[r]& \CO_X\ar[rr]^{\times\prod_{i=1}^d t_i}&& \CO_X(d)\ar[r]& \CF\ar[r]& 0 }\]
and $\RH^1(X,\CO_X)=0$.

\env{cor}{
For any $t\in P_1\bs\set0$, $t(y)=0,y\in|Y|^\cl$, $C=C_y$, 
\[\fct{P/tP}{\set{f\in C[T]\mid f(0)\in E}}{x\mod tP_{i-1}}{\theta_y(x)T^i}\]
is an isomorphism betweem graded rings, where
\[P/tP=E\oplus\bigoplus_{i\ge 1} P_i/tP_{i-1}.\]
}



\subsection{Vector bundles}
Let $X$ be an integral Dedekind scheme with generic point $\eta$.
Let $\infty\in|X|$ be a closed point.
Let $K=\CO_{X,\eta}$ be the field of rational functions on $X$.
We suppose that $X-\set\infty$ is affine, i.e., $X-\set\infty=\Spec A$ where $A=\\RH^0(X\-\set\infty,\CO_X)$.
Let $t$ be a uniformizer in $\CO_{X,\infty}$.

Denote by $\cBun_X$ the category of vector bundles on $X$, i.e., locally free $\CO_X$-modules of finite rank.

Denote by $\cC$ the category of triples $(M,W,u)$, where $M$ is a projective $A$-module of finite type, $W$ is a free $\wh\CO_{X,\infty}$-module of finite type and
 \[u: M\otimes_A \wh\CO_{X,\infty}[1/t]\simto W[1/t]\]
is an isomorphism.

\env{thm}{
There is an equivalence of categories
\[\begin{split}
\cBun_X&\simto \cC\\
\CE&\longmapsto (\Gamma(X-\set\infty,\CE),\wh\CE_\infty,\can).
\end{split}\]
Here $\can$ is induced by $\Gamma(X-\set\infty,\CE)\inj \CE_\eta=\CE_\infty[1/t]$.
}

Moreover, if $\CE$ corresponds to $(M,W,u)$, then $\Gamma(X,-)$ on $\CE$ has a resolution
\[\Gamma(X,\CE)\to M\oplus W\sto{\partial}W[1/t],\]
where $\partial(m,w)=u(m)-w$. Thus
\[\RH^0(X,\CE)=u(M)\cap W,\qquad
\RH^1(X,\CE)=\frac{W[1/t]}{W+u(M)}.\]

Suppose $X$ is complete. Then there is a map $\deg:|X|\to \BN_+$ such that $\deg(\div(f))=0$. Assume $\deg \infty=1$. Then 
\[\RH^0(X,\CO_X)=\set{f\in K^\times\mid \div(f)\ge 0}\cup\set0=\set{f\in K^\times\mid \div(f)=0}\cup\set0\]
is a field. Denote by $E=\RH^0(X,\CO_X)\subset K$.

Denote by 
\[\deg=-\ord_\infty:A\to \BN\cup\set\infty.\]
Then $E=A^{\deg\le0}=A^{\deg=0}$. Note that 
$A^{\deg\le d}=\RH^0(X,\CO_X(d[\infty]))$ and ths sheaf $\CO_X(d[\infty])$ corresponds $(A,t^{-d}\wh\CO_{X,\infty},\can)$,
\[\RH^1(X,\CO_X(d[\infty]))=\frac{K}{t^{-d}\CO_{X,\infty}+A}.\]
In particular,
\[\RH^1(X,\CO_X(-\infty))=\frac{K}{t\CO_{X,\infty}+A}\]
is zero iff $A$ is {\em Euclidean}, i.e., for any $x,y\in A$ with $y\neq 0$, there is $a\in A$ such that $\deg(x/y-a)<0$. $\RH^1(X,\CO_X)=0$ iff $A$ is {\em almost Euclidean}, i.e., $\deg(x/y-a)\le0$.

Now for our $X$, $\RH^1(X,\CO_X)=0$ but $\RH^1(X,\CO_X(-1))\neq0$, since $B_e$ is almost Euclidean but not Euclidean.


\subsection{Harder-Narasimhan filtrations}
See Yves Andr\'e, {\em Slope filtrations} \url{https://arxiv.org/abs/0812.3921}.

Consider
\begin{itemize}
\item an exact category $\cC$,
\item an abelian category$\cA$,
\item an exact faithful functor $\CF:\cC\to \cA$, called {\em generic fiber functor}, such that for any $X\in\cC$, $\CF$ induces an equivalence between strict sub-objects of $X$ and sub-objects of $\CF(X)$, the inverse functor is called {schematical closure}.
\item an additive map $\rk:\Obj(\cC)\to\BN$, i.e., it factors through $\RK_0(\cC)\to \BZ$, such that $\rk(X)=0$ iff $X=0$,
\item an additive map $\deg:\Obj\cC\to \BR$ such that for $u:X\to Y$, if $\CF(u)$ is an isomorphism, then $\deg X\le \deg Y$ with equality iff $u$ is an isomorphism.
\end{itemize}

\env{exam}{
Let $X$ be a complete integral Dedekind scheme.
Then $k(X)=\CO_{X,\eta}$ is equipped with $\deg:|X|\to \BN_{\ge1}$ such that for any $f\in k(X)^\times,\deg(\div(f))=0$. Take $\cC=\cBun_X,\cA=\cVect_{k(X)}$, $\CF(\CE)=\CE_\eta$. Then the strict sub-objects of $\CE$ are
locally direct factors $\CF\subset\CE$. For any $V\subset \CE_\eta$, $\CE\cap V$ is a strict sub-object of $\CE$.

The degree map induces $\deg:\Pic(X)\to\BZ$ and then $\deg:\cBun_X\to \BZ$ via $\deg(\CE):=\deg(\det\CE)$.
Then if $u:\CE\to\CE'$ induces an isomorphism $\CE_\eta\simto \CE_\eta'$, then
\[0\ra\CE\ra\CE'\ra\CF\ra0\]
 with torsion $\CF$, and $\deg\CE'=\deg\CE+\deg\CF$.
 $\CF$ can be written as $\CF=\oplus i_{x*}M_x$ where $M_x$ is finite length $\CO_{X,x}$-module and
 \[\deg\CF=\sum \length_{\CO_x}(M_x)\deg(x).\] 
}

\env{exam}{
If $\cC=\cA$ is an abelian category, $\CF=\Id$, we require additive maps  $\deg$ and $\rk$, such that $\rk(X)=0$ iff $X=0$.
}

\env{exam}{
Let $k$ be a field, $\cBT_k\otimes\BQ$ is the category of $p$-divisible groups over $k$ up to isogeny. This is an abelian category. We take $\rk$ to be the height and $\deg$ the dimension of associated formal group. Then the Harder-Narasimhan filtration in this category is the slope filtration. For example, 
\[0\ra H^\circ\ra H\ra H^\et\ra 0\]
is part of this filtration.
}

\env{exam}{
Let $L/K$ be an extension. Let $\cC$ be the category of vector spaces $V$ over $K$ with a finitely decreasing fitration on $V_L$. The exactness should be strictly compatible with fltrations. Define  
\[\rk=\dim_K V\qquad \deg=\sum i\cdot \dim \gr^i \Fil V_L.\]
Define $\CF:\cC\to\cVect_K$ to be the forgetful functor. Then the deserved property follows from
\[\deg=N\dim V+\sum_{i<N}\dim \Fil^i V_L,\quad N\ll0.\]
}

\env{exam}{
Let $k$ be a perfect field with characteristic $p$,
$\sigma$ the Frobenius on $K_0=W(k)_\BQ$. Let $K/K_0$ be a finite ramified extension. Denote by $\varphi$-$\cModFil_{K/K_0}$ the category of $(D,\varphi,\Fil D_K)$ where $(D,\varphi)$ is an isocrystal.
Denote by $\rk=\dim_{K_0}D,\deg=t_H-t_N$.
Then semi-stable slope $0$ objects are weakly admissible filtered isocrystals.
}

\env{exam}{
Let $\CR$ be a Bezout ring, $\CE\subset \CR$ a field with a nontrivial valuation $v:\CE\to\BR\cup\set{-\infty}$.
Let $\sigma$ be an endomorphism that stabilizes $\CE$ such that $v(\sigma(x))=v(x)$. We assume that $\CE^\times=\CR^\times$ and for any nonzero $x\in\CR$ such that $x^{\sigma-1}\in\CE^\times$, $v(x^{\sigma-1})\ge0$.
Denote by $\cC$ the category of $(M,\varphi)$, where $M$ is a free $\CR$-module with finite rank, $\varphi$ is a $\sigma$-semilinear endomorphism on $M$ such that $\varphi\otimes \Id:M^{(\sigma)}\simto M$. Denote $\CF(M,\varphi)=(M\otimes_\CR\Fr \CR,\varphi\otimes\sigma)$, $\rk=\rk_\CR(M),\deg=-v(\det\varphi)=-v(a)$, where $\det(M,\varphi)=\CR e,\varphi e=ae$.
}

Denote by \[\mu:=\frac{\deg}{\rk}.\]
From now on in this subsection, $X\subseteq Y$ means a strictly sub-object, thus
\[0\ra X\ra Y\ra Y/X\ra 0\]
is exact.
\env{defn}{
$X\in\cC$ is called {\em semi-stable} if for any nonzero strictly sub-object $C'\subset X$, $\mu(X')\le \mu(X)$.
}
\env{remark}{
Any morphism in $\cC$ has a kernel and coker. The kernel of $f:X\to Y$ is the schematical closure of $\ker(\CF(f))$.
}

\env{thm}{
For any nonzero $X\in\cC$, there is a unique filtration 
\[0=X_0\subsetneq X_1\subsetneq \cdots\subsetneq X_n=X\]
such that $X_i/X_{i-1}$ is semi-simple and 
\[\mu(X_1/X_0)>\cdots>\mu(X_{n}/X_{n-1}).\]
}
Define the Harder-Narasimhan polygon $\HN(X)$ to be the concave polygon defined on $[0,\rk X]$ with breaking points $(\rk X_i,\deg X_i)$, i.e., on $[\rk X_i,\rk X_{i+1}]$, it has slope $\mu(X_{i+1}/X_i).$

\env{thm}{
For any $Y\subseteq X$, $(\rk Y,\deg Y)$ is under $\HN(X)$. Thus $\HN(X)$ is the concave hull of $(\rk Y,\deg Y)$ for all $Y\subseteq X$.
}

\env{thm}{
The subcategory $\cC_\lambda^\rss$ of slope $\lambda$ semi-simple objects. is an abelian category, stable under extensions in $\cC$. Thus the Harder-Narasimhan filtrations give a d\'evissage of $\cC$ in $(\cC_\lambda^\rss)_{\lambda\in\BR}$.
}

\begin{proof}[Proof of existance]
If 
\[0\ra X'\ra X\ra X''\ra 0\]
is exact, then
\[\mu(X)=\frac{\rk X'}{\rk X}\mu(X')+\frac{\rk X''}{\rk X}\mu(X'')\in[\mu(X'),\mu(X'')].\]
Here $[a,b]:=[b,a]$ if $a>b$, i.e., the convex hull $\Conv(a,b)$.

If
\[0=X_0\subsetneq X_1\subsetneq \cdots\subsetneq X_n=X\]
is a Harder-Narasimhan filtration of $X$, then
\[\mu(X)\in \Conv(\mu(X_i/X_{i-1}))_{1\le i\le n}.\]
Thus
\[\inf\set{\mu(X_i/X_{i-1}}\le \mu(X)\le \sup\set{\mu(X_i/X_{i-1}}.\]

For nonzero $X$ in $\cC$, consider the condition
\begin{equation}
\text{$Y\subseteq X$ semi-stable and for any $Y'\subsetneq Y\subset X$, $\mu(Y')\le \mu(Y)$,} \tag{*}
\end{equation}
i.e., $Y$ is maximal semi-stable sub-object of $X$. This is equivalent to say, any nonzero $Y''\subset X/Y, \mu(Y'')<\mu(Y)$. In fact, if $Y''=Y'/Y, Y\subsetneq Y'\subset X$, $\mu(Y')\in(\mu(Y),\mu(Y''))$ and thus $\mu(Y'')<\mu(Y)$.

\env{lemma}{
At most one $Y\subseteq X$ satisfying (*).
}
Assume $Y_1,Y_2$ satisfy (*). Suppose $Y_1\not\subseteq Y_2$, consider
\[\xymatrix{
\Ker f\ar[r]& Y_1\ar[rr]^f\ar[rd]&&X/Y_2\\
&&\Im f\ar[ru]^\subseteq
}\]
$Y_1/\Ker f\to \Im f$ is an isomorphism in generic fibers, thus $\mu(Y_1/\Ker f)\le \mu(\Im f)$. But $Y_1$ is semi-stable, $\mu(\Ker f)\le \mu(Y_1)\le \mu(Y_1/\Ker f)\le \mu(\Im f)<\mu(Y_2)$. By symmetric, $\mu(Y_2)<\mu(Y_1)$ if $Y_2\not\subset Y_1$. Thus $Y_1\subseteq Y_2$ or $Y_2\subseteq Y_1$.

\env{lem}{
$\mu_{\max}(X):=\sup\set{\mu(Y)\mid 0\neq Y\subset X}<+\infty.$
}
Take
\[0=X_0\subsetneq \cdots\subsetneq X_n=X\]
such that $0=\CF(X_0)\subsetneq\cdots\subsetneq \CF(X_n)=\CF(X)$ is a Jordan-H\"older filtration.
For nonzero $Y\subseteq X$, take $0=Y_0\subseteq\cdots\subseteq Y_n=Y$ such that $\CF(Y_i)=\CF(Y)\cap\CF(X_i)$.
Consider $u_i:Y_i/Y_{i-1}\inj X_i/X_{i-1}$, $\CF(u_i):\CF(Y_i/Y_{i-1})\inj \CF(X_i/X_{i-1})$. Since  $\CF(X_i/X_{i-1})$ is simple, $Y_i=Y_{i-1}$ or $\CF(u_i)$ is an isomorphism, thus $\mu(Y_i/Y_{i-1})\le \mu(X_i/X_{i-1})$ and then  $\mu(Y)\le \sup\set{\mu(Y_i/Y_{i-1})}\le \sup\set{\mu(X_i/X_{i-1})}.$

\env{lem}{$\mu_{\max}(X)$ is reached.}
It's clear if $\deg:\cC\to \BZ$.

Now we take $Y$ such that $\mu(Y)=\mu_{\max}(X)$ with maximal rank, then $Y$ satisfies (*).

Let's back to the proof. Set $X_1\subset X$ satisfying (*) and $X_i/X_{i-1}\subset X/X_{i-1}$ satisfying (*) inductively. The existance then follows.

If we have such a filtration, then $X_1\subset X$ satisfying (*). In fact, for $Y\subset X/X_1$, $0=Y_1\subset Y_2\subset\cdots\subset Y_n=Y$ such that $v_i:Y_i/Y_{i-1}\inj X_i/X_{i-1}$. Then $\mu(Y_i/Y_{i-1})\le\mu(\Im v_i)\le \mu(X_i/X_{i-1})$ and $\mu(Y)\le \sup\set{\mu(Y_i/Y_{i-1})}\le \sup\set{\mu(X_i/X_{i-1})}=\mu(X_1/X_0)$. The uniqueness then follows by induction.
\end{proof}

\section{Classification of vector bundles}
Assume $E/\BQ_p$, $F/\BF_q$ is algebraically closed. Let $X_E/\Spec E$ be the Fontaine-Fargues curve.

\envn{thm}{GAGA, Kedlaya-Liu}{
There is an equivalence of categories
\[\cCoh_X\simto \cCoh_{X^\an}.\]
}
\subsection{Construction of some vector bundles}
Recall $X_E=\Proj(P_{E,\pi})$. Denote by $\CO_{X_E}(d)$ the module with respect to the graded $P_{E,\pi}$-module $P_{E,\pi}[d]$. This is a line bundle on $X_E$.

\env{remark}{
$X_E$ does not depend canonically on the choice of $\pi$, but $\CO_{X_E}(1)$ does: another choice of uniformizing element leads to an isomorphic line bundle but the isomorphism is not canonical.
}

Since $X$ is ``complete'', $\deg(\div f)=0$, we have
\[\deg:\Pic(X_E)=\Div(X_E)/\div(E(X_E)^\times)\to \BZ.\]
Define $\deg(\CE)=\deg(\det\CE)$ for vector bundle $\CE$.
Take $\mu=\deg/\rk$, we get Harder-Narasimhan reduction theory.

\env{prop}{
We have an isomorphism $\deg:\Pic(X_E)\simto \BZ$, i.e. $\Pic(X_E)=\pair{\CO_{X_E}(1)}$.
}
This is a consequence of $X_E-\set\infty$ is affine and the ring of global sections are PID.

For $E'/E$, $X_E':=X_E\otimes_E E'$.
If $E_h/E$ is unramified of degree $h$, then $\varphi_{E_h}=\varphi_E^h,W_{\CO_{E_h}}=W_{\CO_E}$. Replacing $E$ by $E_h$ does not change $Y_{E_h}=Y_E$, it changes the Frobenius.
\[\xymatrix{
X^\ad_{E_h}\ar[d]_{\BZ/n\BZ}\ar@{=}[r]& Y^\ad/\varp^{n\BZ}\ar[d]^{\pi_h}\\
X_E^\ad\ar@{=}[r]&Y^\ad/\varp^\BZ. 
}\]
Then by GAGA, we get a $\BZ/h\BZ$ Galois cover
\[\xymatrix{X_{E_h}\ar[d]^{\pi_h}\\ X_E.}\]
Thus
\[\xymatrix{
(X_{E_h})_{h\ge 1}\ar[d]\\ X_E
}\]
is a $\wh\BZ$-pro Galois cover.

We have $\pi_{E_h}^*\CO_{X_E}(d)=\CO_{X_{E_h}}(hd).$

\env{defn}{
For any $\lambda=d/h\in \BQ,(d,h)=1,h>0$, define
\[\CO_{X_E}(\lambda)=\pi_{h}* \CO_{X_{E_h}}(d).\]
}
It's of rank $h$ and degree $d$. It's semi-stable of slope $\lambda$ since pushforwards of a semi-stable vector bundle by a finite \etale\ Galois cover are still semi-stable.

We have
\[\CO(\lambda)\otimes\CO(\mu)=\bigoplus_{\text{finite}} \CO(\lambda+\mu),\]
\[\CO(\lambda)^\vee=\CO(-\lambda).\]
\[\Hom(\CO(\lambda),\CO(\mu))=\bigoplus_{\text{finite}}\RH^0(X,\CO(\mu-\lambda)\]
is zero if $\lambda>\mu$ since $\RH^0(X_E,\CO(\frac{d}{h}))=\RH^0(X_{E_h},\CO_{X_{E_h}}(d))=0$ if $d<0$.
\[\Ext^1(\CO(\lambda),\CO(\mu))=\bigoplus\limits_{\text{finite}} \RH^1(X,\CO(\mu-\lambda))\]
is zero if $\lambda\le \mu$ since $\RH^1(X_E,\CO(\frac{d}{h}))=\RH^1(X_{E_h},\CO_{X_{E_h}}(d))=0$ if $d\ge0$.

\env{thm}{
(1) Any slope $\lambda$ semi-stable vector bundle is isomorphic to a direct sum of $\CO_X(\lambda)$.

(2) The Harder-Narasimhan filtration of a vector bundle is split.

(3) There is a bijection between
\[\set{\lambda_1\ge\cdots\ge \lambda_n\mid \lambda_i\in\BQ,n\in\BN}\]
and the isomorphic classes of vector bundles on $X$ as
\[(\lambda_i)\longmapsto \left[\bigoplus_i\CO(\lambda_i)\right].\]
}

\env{remark}{
(1)+(2)$\iff$(3). Moreover, (1)$\Rightarrow$(2) via the computation of $\Ext^1(\CO(\lambda),$ $\CO(\mu))=0$ if $\lambda\le \mu$. 
}
In particular, denote by $\cBun_X^{\ss,0}$ the abelian category of slope $0$ semi-stable vector bundles over $X$. Then we have an equivalence of categories
\[\begin{split}
\cVect_E&\simto \Bun_X^{\ss,0}\\
V&\mapsto V\otimes_E\CO_X\\
\RH^0(X,\CE)&\mapsfrom \CE.
\end{split}\]
That's to say, a vector bundle over $X$ is trivial iff it's semo-stable of slope $0$.

More generally, $\End(\CO(\lambda))=D_\lambda^\op$, where $D_\lambda$ is the division algebra over $E$ with invariant $\lambda$. We have an equivalence of categories
\[\begin{split}
\cVect_{D_\lambda}&\simto \Bun_X^{\ss,\lambda}\\
V&\mapsto V\otimes_{D_\lambda}\CO(\lambda)
\end{split}\]

\subsection{From isocrystals to vector bundles}
Denote by $\vE=\wh{E^\ur}$ endowed with Frobenius $\sigma$.
Denote by $\cphimod_{\vE}$ the abelian category of isocrystals, which is semi-stable by Dieudonn\'e-Mannin.
\[\cphimod_{\vE}=\bigoplus_{\lambda\in\BQ}\cphimod_{\vE}^\lambda.\]
For any $\lambda$, there is a unique simple object $N_\lambda=\pair{e,\varphi(e),\dots,\varphi^{h-1}(e)},\lambda=d/h$ with $\varpi^h(e)=\pi^de$.

We have a $\otimes$-exact functor 
\[\fct{\cphimod_{\vE}}{\cBun_X}{(D,\varp)}{\CE(D,\varp)}\]
where $\CE(D,\varp)$ is the module associated to the graded $P$-module
\[\bigoplus_{d\ge 0}(D\otimes_E B)^{\varphi\otimes\varphi=\pi^d}.\]
Via GAGA, $\CE(D,\varphi)^\ad$ is a vector bundle on $Y/\varphi^\BZ$ corresponding to the $\varphi$-equivariant vector bundle $(D\otimes_{\breve E}\CO_Y,\varp\otimes\varp)$.

If $(D,\varphi)$ is simple of slope $\lambda$, then $\CE(D,\varphi)=\CO_X(-\lambda).$
Thus via Dieudonn\'e-Manin classification theorem, this functor is essentially surjective.

\section{Periods of $p$-divisible groups}
The main tool is the classification theorem. Take $E=\BQ_p$ to simplify. Let $C/\BQ_p$ be an algebraically closed field with $C^\flat=F$. Thus there is $\infty\in|X|$ with $k(\infty)=C$.

Denote by $\cBT_{\CO_C}$ the category of Barsotti-Tate $p$-divisible groups over $\CO_C$. We want to explain the functor
\[\fct{\cBT_{\CO_C}}{\set{\text{Modifications of vector bundles}}}{H}{[0\ra V_p(G)\otimes\CO_X\ra\CE_H\ra i_{\infty*}\Lie H[\frac{1}{p}]\ra 0]}\]
where $V_p(H)\otimes \CO_X$ is a trivial vector bundle with fiber $V_p(H)$, $\CE_H=\CE(D,p^{-1}\varphi)$ is a covariant isocrystal of the reduction of $H$.

\subsection{Periods in characteristic $p$}
Let $k/\BF_p$ be a perfect field. A Dieudonn\'e crystal is a free $W(k)$-module of finite rank with endomorphisms $F,V$, where $F$ is $\sigma$-linear, $V$ is $\sigma^{-1}$-linear, $FV=VF=p$. Then 
\[\begin{split}
\cBT_k&\simto \set{\text{Dieudonn\'e crystals}}\\
H&\mapsto \BD(H).
\end{split}\]

\subsection{The covectors}
Denote by 
\[W_n=\set{[x_0,\dots,x_{n-1}]}=W/V^n W\]
the ring of trucated Witt vectors of length $n$. It's an affinte unipotent group scheme, isomorphic to $\BA_k^n$.
We have
\[\xymatrix{
W_n\ar@{^(->}[r]^V&
W_{n+1}\ar@{^(->}[r]^V&
W_{n+2}\ar@{^(->}[r]&\cdots
}\]
where $V([x_0,\dots,x_{n-1}])=[0,x_0,\dots,x_{n-1}]$.

Denote by 
\[\CWt^u:=\ilim_{n\ge 1}W_n=\set{[x_n]_{n\le 0}\mid x_n=0\text{ for }n\ll0}.\]
the ring of unipotent Witt covectors.
Here
\[[x_n]+[y_n]=[z_n]\]
with $x_n=P_k(x_{n-k},\dots,x_n,y_{n-k},\dots,y_n),k\gg0$, $P_k$ is the polynomial gives the addition of Witt vectors
\[\sum_{n\ge 0} V^n[x_n]+\sum_{n\ge 0} V^n[y_n]=\sum_{n\ge0}V^n[P_n(x_0,\dots,x_n,y_0,\dots,y_n)].\]

The problem of this ring is  $\Hom(\mu_p,\CWt^u)=0$ since $\mu_p$ is not unipotent. So we need Fontaine's Witt covectors.
Let $R$ be an $\BF_p$-algebra,
\[\CWt(R):=\set{[x_n]\mid x_n\in R, (x_n)_{n\le N}\text{ nilpotent }N\ll0}.\]
It's well-define, i.e., for any $n$, the sequence
\[(P_k(x_{n-k,\dots,x_n,y_{n-k},\dots,y_n}))_{k\ge0}\]
is constant for $k\gg0$.

We have $F[x_n]=[x_n^p], V[\dots,x_{-1},x_0]=[\dots,x_{-2},x_{-1}]$.
For $H\in\cBT_k$,
\[\BD(H)=\Hom_k(H,\CWt_k).\]
It's some kind of Pontryagin duality. The action of $F,V$ via them on $\CWt$. Then if $M=\BD(H)$, one finds back $H$ via
\[H=\Hom_{F,V}(M,\CWt).\]


\env{example}{
$M=W(k)\cdot e$, $Fe=e,Ve=pe$, $R$ is an $\BF_p$-algebra.
\[\Hom_{F,V}(M,\CWt(R))=\set{[x_n]_{n\le 0}\mid x_n\in R, x_n^p=x_n, \sum_{n\le N} Rx_n \text{ nilpotent }, N\ll 0}.\]
Thus $x_n=0$ for $n\ll 0$ and 
\[\Hom_{F,V}(M,\CWt(R))=\BQ_p/\BZ_p(R).\]
This means $M=\BD(\BQ_p/\BZ_p)$, $\BQ_p/\BZ_p=\set{[x_n]_{n\le 0}\in\CWt\mid x_n^p=x_n}.$
}

\env{example}{
$M=W(k)\cdot e, Fe=pe,Ve=e$,
\[\Hom_{F,V}(M,\CWt(R))=\set{[x_n]_{n\le 0}\mid x_n\in R, x_{n-1}=x_n, x_n \text{ nilpotent }}=\hGm(R).\]
Then $M=\BD(\hGm)$, $\hGm\simto \CWt^{V=\Id},x\mapsto \sum_{n\le 0}V^n[x]$.
}

\env{example}{
Let $\lambda=d/h\in(0,1),d\ge 1,(d,h)=1$. Denote 
\[\begin{split}
H_\lambda
&=\Ker(V^d-F^{h-d}:\CWt\to \CWt)\\
&=\set{[\dots,z_{d-1}^{p^{h-d}},\dots,z_1^{p^{h-d}},z_{d-1},\dots,z_1]\in\CWt\mid z_1,\dots,z_{d-1}\text{ nilpotent}}
\end{split}\]
the formal $p$-divisible group of slope $\lambda$. Then $H_\lambda=\Spf(k[[z_0,\dots,z_{d-1}])$. Denote by $M_\lambda=\BD(H_\lambda)$. Then $(M_\lambda[\frac{1}{p}],F)$ is a simple isocrystal of slope $\lambda$.

If $[x_k]_{k\ge 0}+[y_k]_{k\ge0}=[p_k(x_0,\dots,x_k,y_0,\dots,y_k)]_{k\ge0},$ then 
\[(x_0,\dots,x_{-d+1})+_{H_\lambda}(y_0,\dots,y_{-d+1})=(z_0,\dots,z_{-d+1}),\]
\[z_0=\lim_{k\to+\infty} P_{kd}(x_{d-1}^{p^{k(h-d)}},\dots,x_0^{p^{k(h-d)}},\dots,x_{-d+1},\dots,x_0,\dots,y_0)\]
for the $(x_0,\dots,x_{-d+1},y_0,\dots)$-adic topology on $k[[x_i,y_i]]$.
}

\subsection{Period isomorphism in characteristic $p$}
Let $F/\ov\BF_p$ be  a perfectoid field, $H$ a $p$-divisible formal group over $\ov\BF_p$. Let $M=\BD(H)$ be the contravariant Dieudonn\'e module. Denote
\[\BWt=\plim_{V} \CWt=\set{[x_n]_{n\in\BZ}\mid (x_n)_{n\le N}\text{ is nilpotent},N\ll 0}.\]
Then 
\[0\ra W\ra \BWt\ra \CWt\ra 0\]
is exact.

Since
\[H(\CO_F)=\Hom(\Spf\CO_F,H)=\plim_{(0)\neq \fa\subset\CO_F} H(\CO_F/\fa),\]
\[\CWt(\CO_F)=\plim \CWt(\CO_F/\fa)=\set{[x_n]_{n\le0}\mid x\in\CO_F,\limsup_{n\to-\infty}|x_n|<1}.\]
We have 
\[H(\CO_F)=\Hom_{W(k)[F,V]}(M,\CWt(\CO_F)),\]
$H$ is formal if and only if $F$ is topologically nilpotent on $M$ and $\CO_F$ is perfect.

\env{prop}{
The projection $\BWt(\CO_F)\surj\CWt(\CO_F)$ induces 
\[\Hom_{W(k)[F,V]}(M,\BWt(\CO_F))\simto \Hom_{W(k)[F,V]}(M,\CWt(\CO_F)).\]
An inverse is given by 
\[u\mapsto[x\mapsto\lim_{k\to+\infty} F^{-k}\wt{u(F^k x)}].\]
}
If $(D,\varphi)=(M[\frac{1}{p}],F)$, one deduces
\[H(\CO_F)=\Hom_\varphi(D,\BWt(\CO_F)).\]
Now
\[\begin{split}
\BWt(\CO_F)&\inj \CO(Y_F)=B_F,\\
V^n[x_n]&\mapsto [x_n^{p^{-n}}] p^n.
\end{split}\]
Thus 
\[\BWt=\set{\sum_{n\in\BZ}[x_n]p^n\mid x_n\in\CO_F,\limsup_{n\to-\infty}|x_n|^{p^n}<1}\subset B_F^+=\CO(Y_F\cup\set{y_{\cris}})\]
contains all periods with slope in $[0,1]$.

\env{prop}{
$\Hom_\varphi(D,\BWt(\CO_F))=\Hom_\varphi(D,B_F).$
}

\env{example}{
For $\lambda=d/h\in(0,1]$, 
\[\begin{split}
H_\lambda(\CO_F)&=B_F^{\varphi^h=p^d}=\BWt(\CO_F)^{V^d=F^{h-d}}\\
&=\set{\sum_{k=0}^{d-1} \sum_{n\in\BZ} [x_k^{p^{-nh}}]p^{nd+k}\mid x_0,\dots,x_{d-1}\in\fm_F}.
\end{split}\]

If $\lambda=1$, we have an isomorphism
\[\begin{split}
\fm_F&\simto B^{\varphi=p}\\
\varepsilon&\mapsto \sum_{n\in\BZ} [\varepsilon^{p^{-n}}]p^n.
\end{split}\]
Denote by 
\[\CL=\sum_{n\ge0}\frac{T^{p^n}}{p^n}\in\BQ_p[[T]]\]
the logarithm of a $p$-typical formal group law $\CF/\BZ_p$.
Then 
\[X+_\CF Y=\CL^{-1}(\CL(X)+\CL(Y))\in\BZ_p[[X,Y]].\]
For $X+_{\hGm} Y=XY+X+Y, \log_{\hGm}=\log(1+T)$.
Denote by $E(T)=\exp(\CL(T))\in\BZ_p[[T]]$ the Artin-Hasse map. Then $E:\CF\simto \hGm$ and we have a commutative diagram
\[\xymatrix{
(\fm_F,+_{\CF})\ar[rrr]^\sim_{\varepsilon\mapsto \suml_{n\in\BZ}[\varepsilon^{p^{-n}}]p^n} \ar[d]^{\simeq}_E &&& B^{\varphi=p}\ar@{=}[d]\\
(\fm_F,+_{\hGm})\ar[rrr]^\sim_{\varepsilon\mapsto \log([1+\varepsilon])=t} &&&B^{\varphi=p}.
}\]

If $\lambda=d/h\notin[0,1]$, $B^{\varphi^h=p^d}$ has no explicit description: the Banach-Colmez space $\BB^{\varphi^h=p^d}$ is not representatble by a pefectoid space but by a diamond (algebraic space for pro-\etale\ topology).
}

\subsection{Periods in unequal characteristic}
Let $C/\BQ_p$ be an algebraically closed field, $F=C^\flat$,
$H/\CO_C$ a formal $p$-divisible group.
We are going to look at the universal cover $\plim_{\times p}H$ of $H$.

\env{prop}{
There is an isomorphism $\plim_{\times p}H(\CO_C)\simto \plim_{\times p}H(\CO_C/p\CO_C)$. The inverse is given by sending $(x_n)_{n\ge0}$ to $(\lim\limits_{k\to+\infty} p^{-k}\wt x_{n+k})_{n\ge 0}$ via any lift of $H(\CO_C)=\plim_{\times p}H(\CO_C/p^i\CO_C)\to H(\CO_C/p\CO_C).$
}
The last isomorphism comes from that $H$ is $p$-divisible $p^\infty$-torsion, $H_\eta=\circB^d_C$, while $\times p$ contracts everything to $0$.

Suppose $\BH/\ov\BF_p$ is a $p$-divisible group with an identification
\[\BH\otimes_{\ov\BF_p}\CO_C/p\CO_C\simto H\otimes_{\CO_C}\CO_C/p\CO_C.\]
Take $\varpi^\sharp=p$, then
\[\begin{split}
\plim_{\times p} H(\CO_C)
&=\plim_{\times p}H(\CO_C/p\CO_C)=\plim_{\times p}\BH(\CO_C/p\CO_C)=\plim_{\times p}\BH(\CO_F/\varpi\CO_F)\\
&=\plim_{\times p}\BH(\CO_F)=\BH(\CO_F)=\Hom_\varphi(D,B_F),\end{split}\]
where $(D,\varphi)=\BD(\BH)$.

\env{remark}{
More generally
\[\plim_{\times p}H_\eta=\circB^{d,1/p^\infty}_C\]
is a pre-perfectoid ball $\Spf[[X_0^{1/p^\infty},\dots,X_{d-1}^{1/p^\infty}]]_\eta$ over $C$, where $H_\eta=\circB^d_C$. The tilt of this is $(\BH^{1/p^\infty}\otimes_{\ov\BF_p}\CO_F)_\eta$.
}
Let 
\[\log_H:H_\eta\to\Lie H\otimes_{\CO_C} \Ga^\rig\]
be the logarithm of the formal group $H_\eta$. This is a morphism of rigid analytic groups, which is an \etale\ $H(\CO_C)[p^\infty]$-tower.

By applying $\plim_{\times p}$ on the exact sequence
\[0\ra H(\CO_C)[p^\infty]\to H_\eta\sto{\log_H} \Lie H\otimes_{\CO_C}\Ga^\rig\ra 0,\]
we get
\[0\ra V_p(H)\to \plim_{\times p}H(\CO_C)\sto{\log_H(x_0)} \Lie H[\frac{1}{p}]\ra 0.\]
Rewrite it in terms of covariant isocrystals, we get
\[0\ra V_p(H)\to (D\otimes_{\vQ_p}B_F)^{\varphi=p}\ra \Lie H[\frac{1}{p}]\ra 0.\]
Here let $\Fil D_C=\omega_{H^D}[\frac{1}{p}]\subset D_C$ be the Hodge filtration. Then $D_C/\Fil D_C=\Lie H[\frac{1}{p}]$ and the last map in the exact sequence is given by
\[\xymatrix{
(D\otimes_{\BQ_p}B_F)^{\varphi=\Id}\ar[r]\ar@{^(->}[d]&\Lie H[\frac{1}{p}]\\
D\otimes_{\vQ_p}B_F\ar@{->>}[r]^-{\id\otimes\theta}&D_C\ar@{->>}[u]
}\]

\env{example}{
When $H=\hGm$, this is just the fundamental exact sequence.
}

\env{prop}{
$V_p(H)\ra(D\otimes B)^{\varphi=p}$ induces an isomorphism
\[V_p(H)\otimes_{\BQ_p}B[\frac{1}{p}]^{\varphi=\Id}\simto 
(D\otimes_{\vQ_p}B[\frac{1}{t}])^{\varphi=\Id}.\]
}
Use \Poincare\ duality, we get a perfect pairing
\[\xymatrix{
V_p(H)\times V_p(H^D)\ar[d]\ar[r]^-\cup
&\BQ_p(1)=\BQ_p t\ar@{^(->}[d]\\
(D\otimes B)^{\varphi=p}\times(D^*\otimes B)^{\varphi=\Id}\ar[r]^-\cup
&B^{\varphi=p}.
}\]
The right hand side map is an isomorphism after inverting   $t$.

\env{coro}{
For any $p$-divisible group $H/\CO_C$, the corresponding $(D,\varphi,\Fil D_C)$ defines a modification of vector bundles on $X_F$ at $\infty\in|X_F|$,
\[0\ra V_p(H)\otimes_{\BQ_p}\CO_X\ra \CE(D,p^{-1}\varphi)\ra i_{\infty*}\Lie H[\frac{1}{p}]\ra 0.\]
In particular, via $D_C=\CE(D,p^{-1}\varphi)_{\infty}\otimes k(\infty)$, $u:\CE(D,p^{-1}\varphi)\surj i_{\infty*}D_C$,  $u^{-1}(i_{\infty*}\Fil D_C)$ is a trivial bundle.
}

\section{Topics on classification theorem}
\subsection{Lubin-Tate space}
Let $\BH$ be a $1$-dimensional hegith $n$ formal $p$-divisible group. Let 
\[\fX=\Def(\BH)\simeq \Spf(W(\ov\BF_p)[[X_1,\dots,X_{n-1}]]).\]
Then we have Gross-Hopkins period morphism, which is an anlog of Griffiths period morphism.
\[\xymatrix{
\qquad\qquad\fX_\eta=\circB^{n-1}_{\vQ_p}\ar[d]^{\pi_\dR} \\
\quad\BP^{n-1}_{\vQ_p}
}\]
Denote $(D,\varphi)=\BD(\BH)$. Then for $x\in\fX(\CO_C)=\fX_\eta(C)$, $\pi_\dR(x)=\Fil D_C\subset D_C$ is a codimension $1=\dim\BH$ subspace, that is, the Hodge filtration of $x^* H^\univ/\CO_C$, where $H^\univ/\fX$ is a universal deformation.

\envn{thm}{Lafaille, Gross-Hopkins}{
$\pi_\dR$ is a surjective \etale\ cover.
}
That's to say, any codimension one subspace $\Fil D_C$ is the Hodge filtration of a lift of $\BH$ to $\CO_C$.
This is a $p$-adic analog of Kodaira-Spencer map.
The \'etaleness follows from Grothendieck-Messing deformation theory. 

We have $\CE(D,p^{-1}\varphi)=\CO_X(\frac{1}{n})$.
\env{coro}{
For any degree $1$ modification of $\CO_X(\frac{1}{n})$,
\[0\ra \CE\ra\CO_X(\frac{1}{n})\ra \CF\ra0\]
where $\CF$ is a degree $1$ torsion coherent sheaf, we have trivial $\CE\simeq\CO_X^n$.
}

Conversely,
\env{prop}{
For 
\[0\ra\CO_X^n\ra \CE\ra \CF\ra 0\]
where $\CF$ is a degree $1$ torsion coherent sheaf, we have $\CE=\CO_X(\frac{1}{d})\oplus\CO_X^{n-d}, 1\le d\le n$.
}
The modification is given by a surjection
\[u:C(-1)^n=(t^{-1}B_\dR^+/B_\dR^+)^n\surj L,\]
where $L$ is a one-dimensional $C$-vector space. Here $\wh \CO_{X,\infty}^n\subset\wh\CE_\infty\subset t^{-1}\wh \CO_{X,\infty}^n$. Up to replacing $\CO_X^n$ by $\CO_X^{n-i}$ and $\CE$ by $\CE'$ with $\CE=\CE'\oplus\CO_X^i$, one can suppose $u:\BQ_p(-1)^n\inj L$, i.e., $u\in\Omega(C)\subset \BP^{n-1}(C)$.

We want to prove this if $u\in\Omega(C)$, then $\CE\cong \CO_X(\frac{1}{n})$. Let $D=\End(\CO_X(\frac{1}{n}))=D_{\frac{1}{n}}$ be the division algebra with invariant $\frac{1}{n}$. It induces $D\otimes_{\BQ_p}\CO_X\simto \ul\End(\CO_X(\frac{1}{n}))$ and $D_X^{\op,\times}\simto \ul\Aut(\CO_X(-\frac{1}{n}))=\ul\GL(\CO_X(-\frac{1}{n}))$ as $X$-group schemes.
Thus $(D^\op)^\times_X$-torsors over $X$ (pure inner form of $\GL_n$) is equivalent to $\GL_n$-torsors on $X$ (vector bundle of rank $n$).
In fact, if $\CT$ is a topos, $G$ is a group on $\CT$, $\BT$ is a $G$-torsor in $\CT$, $H=G^\BT$ is the inner twisting of $G$,
\[[\BT]\in\RH^1(\CT,G)\ra\RH^1(\CT,G_\ad)\ni[H]=[\ul\Aut(\BT)].\]
Then $t\mapsto \ul\Isom(\BT,t)$ induces the equivalence between $G$-torsors and $H$-torsors.

Now
\[0\ra\CO_X^n\ra \CE\ra \CF\ra 0\]
is equivalent to
\[0\ra \CO_X(-\frac{1}{n})\ra \CE'\ra \CF'\ra 0\]
as $D^\op\otimes\CO_X$-module. Take dual modification, we get
\[0\ra \CE''\ra \CO_X(\frac{1}{n})\ra \CF''\ra 0\]
as $D\otimes\CO_X$-module.

\envn{thm}{Drinfeld}{
Any element of $\Omega(C)$ is the Hodge filtration of a special formal $\CO_D$-module.
}
Hence $\CE''\simeq D\otimes_{\BQ_p}\CO_X$. The result follow by applying $\Hom(\CO_X(\frac{1}{n}),-)$.

\subsection{Proof of the classification for rank two vector bundles}
\env{prop}{
Let $\CF$ be a degree one torsion coherent sheaf on $X$.

(1) If
\[0\ra \CE\ra \CO(d_1)\oplus\CO(d_2)\ra \CF\ra 0\]
with $d_1\neq d_2$, $\CE\cong \CO(d_1-1)\oplus\CO(d_2)$ or $\CO(d_1)\oplus\CO(d_2-1)$.

(2) If
\[0\ra \CE\ra \CO(d)\oplus\CO(d)\ra \CF\ra 0,\]
$\CE\cong \CO(d-\half)$ or $\CO(d-1)\oplus\CO(d).$

(3) If 
\[0\ra \CE\ra \CO(d+\half)\ra \CF\ra 0,\]
$\CE\cong\CO(d)^2$.
}
(1) by explicit computation. (2) is a consequence of Lubin-Tate case. (3) is a consequence of Drinfeld case.

Let $\CE$ be a rank $2$ vector bundle on $X$. Then there is 
\[0\ra \CO(d_1)\ra \CE\ra \CO(d_2)\ra 0.\]
If $d_2\le d_1$, $\Ext^1(\CO(d_2),\CO(d_1))=0$ and $\CE=\CO(d_1)\oplus\CO(d_2)$. If $d_2>d_1$,
\[\xymatrix{
0\ar[r]&\CO(d_1)\ar[r]\ar@{^(->}[d]&\CE\ar[r]\ar@{^(->}[d]&\CO(d_2)\ar[r]\ar@{=}[d]&0\\
0\ar[r]&\CO(d_2)\ar[r]&\CE'\ar[r]&\CO(d_2)\ar[r]&0.
}\]
In both cases,
\[0\ra \CE\ra \CO_X(d)^2\ra \CF\to0.\]

Let $\Fil^\bullet$ be a filtration of $\CF$ such that $\gr^i\CF$ is zero or degree one torsion coherent sheaf, $\forall i$.
Take $\Fil^\bullet\CO_X(d)^2=u^{-1}(\Fil^\bullet \CF)$. Then for any $i$, $\Fil^{i+1}(\CO_X(d)^2)$ is $\Fil^i(\CO_X(d)^2)$, or a degree one modification of $\Fil^i(\CO_X(d)^2)$. By induction on $i\in\BZ$, we get $\Fil^i(\CO_X(d)^2)=\CO(k+\half)$ or $\CO(k_1)\oplus\CO(k_2)$.

\subsection{Weakly admissible implies admissible}
Let $K/\BQ_p$ be a discrete valuation field with perfect residue field. Denote $C=\wh{\ov K}, G_K=\Gal(\ov K/K)$, $K_0=W(k_K)_\BQ$, $\sigma$ the Frobenius on $K_0$. Denote by $\cphimodfil_{K/K_0}$ the category of triples $(D,\varphi,\Fil^\bullet D_K)$, where $(D,\varphi)$ is an isocrystal and $\Fil^\bullet$ is a Hodge filtration of $D_K$. Define
\[\begin{split}
t_N&=v_p(\det\varphi)\\
t_H&=\sum i\dim \gr^i D_K.
\end{split}\]
Denote
\[\BV_\cris(D,\varphi,\Fil^\bullet D_K)=\Fil^0(D\otimes_{K_0}B_\cris)^{\varphi=\Id}=\Fil^0(D\otimes_{K_0}B[\frac{1}{t}])^{\varphi=\Id}.\]
There is a $G_K$-action on it.

\env{defn}{
$(D,\varphi,\Fil^\bullet D_K)$ is {\em admissible} if 
\[\dim_{\BQ_p}\BV_\cris(D,\varphi,\Fil^\bullet D_K)=\dim_{K_0}D.\]
}

\env{defn}{
$(D,\varphi,\Fil^\bullet D_K)$ is {\em weakly admissible} if $t_H=t_N$, and for any sub-isocrystal $D'\subset D$, $t_H(D',\varphi|_{D'},D'_K\cap\Fil^\bullet D_K)\le t_N(D',\varphi|_{D'},D'_K\cap\Fil^\bullet D_K).$
}

\envn{thm}{Colmez-Fontaine}{
Weakly admissible is equivalent to admissible.
}
$\Leftarrow$ is easy.

We reinterpretate in terms of semi-stablity. Take $\deg=t_H-t_N,\rk=\dim_{K_0}D,\mu=\deg/rk$, then $\cphimodfil_{K/K_0}^\wa=\cphimodfil_{K/K_0}^{\rss,0}$.

The action on $G_K$ on $X_{C^\flat}$ stablizes $\infty$.
For any $(D,\varphi,\Fil^\bullet D_K)$, $\CE(D,\varphi)$ is a $G_K$-equivariant vector bundle on $X$ and $\Lambda=\Fil^0(D\otimes B_\dR)$ is a lattice in $\wh\CE_\infty[\frac{1}{t}]$. This gives a modification of $\CE$, denoted by $\CE(D,\varphi,\Fil^\bullet D_K)$.
Then
\[\begin{split}
&\deg\CE(D,\varphi,\Fil^\bullet D_K)\\
=&\deg \CE(D,\varphi)+[\Fil^0 D\otimes B_\dR:D\otimes B_\dR^+]-t_N(D,\varphi)\\
=&\deg(D,\varphi,\Fil^\bullet D_K),
\end{split}\]
and $\RH^0(X,\CE(D,\varphi,\Fil^\bullet D_K))=\BV_\cris(D,\varphi,\Fil^\bullet D_K).$

The classification theorem tells that, if $\CE$ is a semi-stable vector bundle of slope $0$, then $\dim_{\BQ_p}\RH^0(X,\CE)=\rk\CE$. Now for $A\in\cphimodfil_{K/K_0}$,
\env{itemize}{
\item $A$ is admissible$\iff$ $\CE(A)$ is semi-stable of slope $0$ and for any sub-bundle $\CE'\subset \CE(A), \mu(\CE')\le 0$;
\item $A$ is weakly admissible $\iff$ $A$ is semi-stable of slope $0$ and for any strict sub-object $B\subset A,\mu(B)\le 0$.
}
\env{prop}{
There is an equivalence between the category of strict subobject of $A$ and $G_K$-equivariant subobject of $\CE(A)$.
}

If $A$ is weakly admissible, the Harder-Narasimhan filtration of $\CE(A)$ is $G_K$-invariant. Thus it comes from a filtration of $A$. Since $A$ is semi-stable, this is the tautological filtration and then $\CE(A)$ is semi-stable, $A$ is admissible.


\end{document}
